Question

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$. $$ \int \sqrt{x} \ln x d x ; \quad u=\ln x, d v=\sqrt{x} d x $$

Name: Problem 2, Chapter 7: Calculus: Early Transcendentals, Metric Edition
Uploaded: 2024-10-09T12:56:06-08:00
Duration: 2 min 6 s
Channel: Gregory Higby
Description: Evaluate the integral using integration by parts with the indicated choices of u and d v. ∫√(x) lnx d x ; u=lnx, d v=√(x) d x

   Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$.
$$
\int \sqrt{x} \ln x d x ; \quad u=\ln x, d v=\sqrt{x} d x
$$

Calculus: Early Transcendentals, Metric Edition

James Stewart,… 9th Edition

Chapter 7, Problem 2

Instant Answer

Solved by Expert Gregory Higby

10/09/2024

Step 1

We are given that $u = \ln x$ and $dv = \sqrt{x} dx$. We need to find $du$ and $v$. To find $du$, differentiate $u$ with respect to $x$: $$ du = \frac{d}{dx} (\ln x) dx = \frac{1}{x} dx $$ To find $v$, integrate $dv$ with respect to $x$: $$ v = \int \sqrt{x} dx Show more…

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Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$. $$ \int \sqrt{x} \ln x d x ; \quad u=\ln x, d v=\sqrt{x} d x $$

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Key Concepts

Integration_by_Parts

A method for integrating the product of two functions, based on the product rule for differentiation. It states that ? u dv = uv - ? v du, allowing one to reduce a complex integral into simpler parts.

Choosing_u_and_dv

A strategy that involves selecting one part of the integrand to differentiate (u), and the remaining part to integrate (dv). The choice is crucial for simplifying the resulting integral, and criteria such as the LIATE rule are often used to guide this selection.

Integration_of_Power_Functions

The process of finding antiderivatives for functions of the form x^n. This is a key step when integrating functions like ?x, which can be rewritten as x^(1/2), and integrated using standard formulas.

Logarithmic_Functions

Understanding the properties and derivatives of logarithmic functions, such as the derivative of ln x being 1/x, which informs the choice of u when applying integration by parts.

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